Railroad wheat transportation markets in the central plains:
modeling with error correction and directed graphs
David A. Bessler
*,1, Stephen W. Fuller
1Department of Agricultural Economics, Texas A&M University, Blocker Building, College Station, TX 77843-2124, USA
Abstract
Time series methods are used to study the dynamics of regional, export-wheat, railroad rates linking seven central US regions to Texas Gulf ports. Research focuses on the extent and nature of regional rate interactions to determine whether regional rail transportation rates are established independently, through interaction and/or dominated by several regional leaders. Analysis is carried out on 1988±1994 public waybill data for seven Business Economic Areas (BEA) located in Kansas, Oklahoma, Texas and Colo-rado: these regions originate virtually all of the eight million tons of hard red winter wheat annually shipped to Texas Gulf ports. Results show all rail transportation markets linked in varying degrees. Some regions are near-independent or highly exogenous regarding rate-setting, while others interact with and to rates established in other regions. Regions that are dominated by a single carrier tend to be more independent and insulated regarding rate-setting. Export rates in regions dominated by the Union Paci®c (UP) generally account for a signi®cant variation in the rates of other regions. Data show the UP to have been aggressive in providing incentives for country elevators to consolidate for purposes of making unit train shipments, thus their presumed in¯uence on rates of other regions. As expected, export rates for regions with sub-stantial storage and transhipment facilities are sensitive to export rates of regions which ship to the tran-shipment facilities. Finally, results suggest that rate-setting in a particular region is in part a function of the dominant railroadÕs management and its aggressiveness, an expected outcome in an oligopolistic market. Ó 1999 Elsevier Science Ltd. All rights reserved.
The Staggers Rail Act of 1980 dramatically altered government policy toward the railroad industry by adopting a market-oriented paradigm that relaxed restrictions on pricing and capital disinvestment. Most evidence suggests the experiment has been a success. Railroads have reduced employment and increased labor productivity, rationalized capital, and lowered real rates while
*Corresponding author. Tel.: +1-409-845-2116; fax: +1-409-862-1563. 1
Bessler and Fuller are Professors of Agricultural Economics at Texas A&M University. Shuang Qin helped gather the data used in the paper.
increasing rates of return and pro®tability (MacDonald and Cavalluzzo, 1996; Wilson, 1994; Friedlaender et al., 1992; Winston et al., 1990; Winston, 1993).
Many economists correctly predicted that railroad deregulation would lower rates for many products. However, an exception to the general accuracy of these forecasts was with regard to bulk products (grain) where it was thought that shippers could expect higher rates (Levin, 1981). It was argued that railroad rates in the Great Plains would increaseex postregulation because of ineective intermodal competition. However, in contrast to expectations, relatively large declines in rates occurred after deregulation. MacDonald (1989) oers, what is perhaps, the most com-prehensive and de®nitive analysis. MacDonald uses 1981±1985 waybill data, in combination with selected control variables and regression methods to show that eective inter-rail competition was introduced into much of the Plains as a result of deregulation. Virtually all studies into the eect of deregulation on Great Plains grain rates analyzed a similar period and regardless of data or method generated ®ndings that corroborate those of MacDonald (Hauser, 1986; Sorenson, 1984; Fuller et al., 1987).
Previous studies into the eect of deregulation on Plains railroad grain rates focused on rate trends and competitive forces aecting rate levels. In contrast, this study examines the dynamic relationships among regional rail transportation rates that link seven central plain wheat pro-duction regions with export facilities on the Texas Gulf. The objective of this research is to de-termine whether export-wheat rates in Plains transportation markets are established independently, through interaction and/or dominated by several regional leaders. Further, be-cause selected railroads have dominant market shares in most regions, the results are expected to oer a perspective on railroad company pricing. This study is of particular interest since the analyzed period (1988±1994) re¯ects important railroad consolidations and possible sophistica-tion in price-setting that may not have been captured by previous studies that focused on the period immediately after deregulation.
Analysis is carried out on public waybill data that identi®es the origin of shipment by Business Economic Area (BEA). Seven BEA regions were found to virtually originate all wheat shipments to Texas Gulf ports; they included two regions that encompassed the western two-thirds of Kansas (northcentral and northwest (Salina, Kansas)), and southcentral and southwest Kansas (Wichita, Kansas)), a region located in northwest and central Oklahoma (Enid, Oklahoma), a region including the northeast quadrant of Colorado (Denver, Colorado), a region including the Texas (Amarillo, Texas) and Oklahoma panhandles and two regions centered about Kansas City and Dallas-Ft. Worth. Each of the Kansas regions and the Oklahoma region ship about 20% of Texas port receipts while the north Texas region (Dallas-Ft. Worth) contributes about 15%. Remaining economic regions shipped about equal portions (8%) to Texas ports. The regions centered about Kansas City and Dallas-Ft. Worth do not produce important quantities of export-destined wheat but, do include important secondary storage facilities that receive from the other ®ve areas.
the region comprising the Texas/Oklahoma panhandles. Based on waybill distance data and knowledge regarding location of major shipping sites, it was estimated that the UP transported about two-thirds of the export-destined wheat in northcentral and northwest Kansas and the Colorado region.
The gathering system of the ATSF railroad was the most extensive of any railroad serving the study region (Fig. 1). During much of the study period, the ATSF included a substantial gathering/mainline system in the south Kansas and Oklahoma regions, with an estimated market share of about 75% (Ruppel et al., 1990). The ATSF has a strong presence in the Texas/ Oklahoma region, a region shared with the Burlington Northern and in the Dallas-Ft. Worth region, a region served by all major carriers. The Burlington Northern system tends to operate on the periphery of the most intensive wheat producing regions (Fig. 1). Regardless, it is esti-mated to have a signi®cant market share in the Texas/Oklahoma region and in the Colorado and Dallas-Ft. Worth regions. The Colorado market is shared with the UP and ATSF, but with the UP presumed to be the dominant carrier. The Kansas City Southern railroad is limited to north/south track extending from Omaha, Nebraska to Texas Gulf ports but is an important carrier of export-destined wheat from the Kansas City market to Texas ports. The SP includes mainline in the Texas/Oklahoma panhandle and south Kansas regions, but lacks direct links to Texas ports, thus its minor role in the transportation of export-destined wheat from the study region.
1. Method of analysis
Our plan is to study the dynamic relations present in observational data on seven regional rate series. Rather than attempt to model the detailed demand and supply structure of (and trans-portation linkages between) each market, we propose to summarize the time series regularities within and among each market. Our ability to capture ceteris paribus conditions embedded in structural demand and supply models on data observed over a seven year period is judged to be weak. Accordingly, we look to summarize the regularities present in these data and attempt to test restrictions de®ned in terms of observable rates rather than the unobservable parameters of de-mand and supply.
waybill sample. The nominal monthly rate data extend from 1988 through 1994, thus 84 rate observations on each region.
A Vector Autoregressive (VAR) representation and, if appropriate, its derivative error cor-rection model, are particularly useful for eciently summarizing dynamic in¯uences and will be applied here. The VAR representation will be entertained as a starting point; however, we expect to see cointegrating (long-run) relations in the data, as rail rates are prices and each geographical location is (potentially) connected by competition by other carriers or other modes (truck).
The maximum likelihood methods of Johansen (1992 and 1995) and Johansen and Juselius (1990) are used in this paper. They consider the cointegration problem as one of reduced rank regression. Letx(t) be the vector of series under investigation, with the following error correction representation:
Dx t lC 1Dx tÿ1 C kÿ1Dx tÿk1 px tÿ1 WDtu t; 1
where
C i ÿp i1 p k for i1;. . .;kÿ1
and
p ÿI ÿp 1 ÿ ÿp k:
Here the p(i) are (p´p) matrices of autoregressive parameters from a VAR in levels ofx(t) of
lag orderk, l a constant, Dt a set of predetermined variables (seasonal dummy variables or
in-tervention dummy variables),Wthe associated parameter(s) on these predetermined variables and
u(t) a white noise innovation term.
Eq. (1) resembles a VAR model in ®rst dierences, except for the presence of the lagged level of
x tÿk. Its associated parameter matrix, p, contains information about the long-run (cointe-grating) relationship between the variables in the vectorx. There are three possibilities of interest: (a) ifpis of full rank, thenx(t) is stationary in levels and a VAR in levels is an appropriate model; (b) if phas zero rank, then it contains no long-run information, and the appropriate model is a VAR in dierences; and (c) if the rank ofpis a positive number,r, and is less thanp(here in our casep7), there exist matrices aand b, with dimensionsp´r, such thatpab0; the latter case, b0x t, is stationary, even though x(t) is not.
In case (c), the long-run equilibrium between the series is summarized by the vectorb. The rate at which each series adjusts to perturbations in this equilibrium is given by theamatrix. Analysis of the elements ofaandb, are fundamental to our understanding of long-run relations present in our data.
2. Results
levels of the log transformed data. The level of each series appears to vary with geographical distance from the port, with Denver generally being the highest rate (lDenver3.00, mile-age1140, wherel indicates the mean log rate in the Denver region over the 1988±1994 period and mileage is the approximate mileage to the Houston Gulf port), followed by Salina
Fig. 2. Log rates for rail transport of wheat from seven hinterland markets, 1988±1994 (log $/ton). Table 1
Likelihood ratio tests on seasonal binary variables and order of lag in a levels VARa
(Seasonal Dummy Variables)
Lag T p-value
0 87.47 0.00
1 117.53 0.00
2 102.48 0.00
3 104.74 0.00
(Lags)
kvs.k+1 T p-value
0 vs. 1 178.23 0.00
1 vs. 2 47.62 0.53
2 vs. 3 50.69 0.41
aTests on seasonal dummy variables are conducted without and with eleven monthly binary variables, at lags zero, one,
(lSalina2.94, mileage800), Wichita (lWichita2.73, mileage700), Amarillo (lAmarillo2.65, mileage690) , Enid (lEnid2.59, mileage600), Kansas City (lKC2.47, mileage850) and Dallas-Ft. Worth (lDFW2.04, mileage300). Kansas City is the exception to our mean rate and is a positive function of distance observation.
While the means of the market rates may be helpful in understanding the overall level of rates, each series is not particularly wedded to its historical mean. Further, rates in each market are not randomly distributed around their individual means, as rates that are above the market mean in periodtappear likely to be followed in time by rates that are also above the market mean (rate in periodt+ 1 will also be highly likely to be above the historical mean). In short, these data appear to have time series attributes. In Table 1 we consider likelihood ratio tests on monthly (seasonal) dummy variables and orders of lags in VARs on levels of log rates in each market. In the ®rst panel of results labeled ``seasonals'' we consider VARs at consecutive lags: no lags,tÿ1, tÿ2, andtÿ3, each with eleven monthly indicator variables (dummy variables). The null hypothesis at each lag is that the eleven dummy variables, as a group have coecient values equal to zero. We strongly reject the null at each length of lag, indicating that the data have seasonal regularities. The second panel of tests presented under the label ``lags'' are likelihood ratio tests on the order of lag in a VAR, given a constant and eleven monthly ``dummy'' variables. Here the null hypothesis is that the coecients at lagk1 are, as a group, equal to zero. (Lutkepohl (1985) has considered tests of this type and suggests a criticalp-value of 0.01 be used to select lags.) Notice here that the coecient matrix at lag one is signi®cantly dierent from zero at a very lowp-value (<0.01), while matrices at lags two and three requirep-values in excess of 0.10 to reject the null hypothesis.
Given the results presented in Table 1 (a VAR of lag order one with seasonal dummy variables), we next consider the number of long-run ``stationary'' relations present in this seven variable data set. Here we consider tests on the number of non-zero eigenvalues present in thep matrix of the error correction representation of the form:
Dx t px tÿ1 WiWiÿ1 u t:
Herep ÿ(Iÿp1 ), wherep1 is the parameter matrix on VAR with one lag ®t with levels data,I the identity matrix, Wi and Wiÿ1 the monthly dummy coecients associated with month i and month iÿ1 corresponding with xtand xtÿ1 in a levels VAR, and u(t) a white noise innovation term. (Notice that we have noC(1)Dx tÿ1term from Eq. (1), as our VAR in levels is of order
k1, so our error correction model will have kÿ10 lags of ®rst dierences.)
Table 2 gives trace test statistics on the rank of p, the number of cointegrating vectors (r) present in this seven variable set. We oer tests for both a constant in the cointegrating vector (T* tests) and for a constant outside the cointegrating vectors (T tests), as we did not test for time trend in our levels VAR (Table 1). Following the sequential testing procedure suggested by Johansen (1992), we see in Table 2 that we have six cointegrating vectors with a constant included in the cointegrating space.
3. Identi®cation
nature of these six long-run relationships. A priori, theory provides expectations on the behavior of rail rates from alternative locations to a common port. One plausible hypothesis, which is consistent with the result we ®nd here, is that for ``p'' markets connected by competition (a free ¯ow of information and the possibility of arbitrage ± in our case it is plausible that grain could be moved by truck from one market to another), we should observe pÿ1 (independent) long-run relationships (z1;z2;. . .;zpÿ1). For example, if we have rates from three ecient markets,r1,r2, and
r3, arbitrage would dictate that two independent vectors hold these three markets together:
Z1
Z2
b11 b21 0 c1 0 b22 b32 c2
r1
r2
r3 1
2
6 6 4
3
7 7 5
:
In our seven market case, we expect to see, as we do, six cointegrating vectors connecting the seven markets. Further, knowledge of six of these rates and competition would be sucient to give long-run information (up to a constant) on the seventh. Actually, Samuelson (1952) derives the pÿ1 restrictions for the case of the price of a commodity in p markets connected by ®xed transportation rates. In our application, transportation to the Texas Gulf is the commodity and each market is connected to the other via rail or truck transportation. His restrictions would result inpÿ1 pairwise equality restrictions and a non-zero constant in the cointegrating vector, the latter to account for dierences in marginal cost related to distance. We explore these below.
The long-run restrictions that the r cointegrating vectors re¯ect such pairwise restrictions (without imposing equality) suggest a just identi®ed cointegrating space (for details on identi®-cation of the cointegration space see Johansen (1995)).
Table 2
Trace tests on order of cointegration on rail rates from seven marketsa
r T C(5%) D T C(5%) D
0 250.52 132.00 R 249.66 123.04 R
61 179.80 101.84 R 178.96 93.92 R
62 128.21 75.74 R 127.41 68.68 R
63 84.20 53.42 R 83.40 47.21 R
64 48.78 34.80 R 48.03 29.38 R
65 22.59 19.99 R 21.84 15.34 R
66 4.76 9.13 F 4.01 3.84 R
aTests are de®ned in Johansen (1992). Brie¯y,
T ÿN Pp
ir1ln 1ÿki;wherekiis theith ordered (from largest to
smallest) eigenvalue of thePmatrix associated with lagged levels in an error correction representation on ®rst dif-ferences of each series,Nis the number of observations used to estimate the error correction model,ris the number of cointegrating vectors being considered andpis the number of series under study (here 7). The critical values (C(5%)
andC(5%)) are 5% levels as reported in Johansen and Juselius (1990). The tests statistics Tand critical values (C(5%)) without the asterisk indicate the tests is on an unrestricted constant (the constant is not in the cointegrating vector). Tests summarized under the headingTandC(5%) have been calculated under the assumption that a constant is in the cointegrating vector. Johansen (1992) suggests a recursive testing procedure, starting ®rst atr0 with the constant in the cointegrating space T, next testr0 and the constant outside of the cointegrating space T. Proceed then to r61 testing the- test ®rst. Stop testing at the ®rst failure to reject F. The headingD Drelates to the ``decision'' to
The estimated error correction model from these just identi®ed restrictions is given in Eq. (3). The xis correspond to each market as follows: x1Kansas City; x2Dallas-Ft. Worth;
x3Amarillo; x4Enid; x5Wichita; x6Salina; and x7Denver. The ®rst matrix on the right-hand side of the equal sign gives the a matrix of Eq. (1), which summarizes the weights with which perturbations in the long-run-equilibrium enter each of the D xis (left-hand side of
the equal sign). The second matrix on the right-hand side of the equal sign gives b0 of Eq. (1), the cointegrating vectors, so that b0x tÿ1 give perturbations in the long-run equilibrium. Finally, the last matrix gives estimated coecients associated with the seasonal dummy vari-ables. Parentheses give t-statistics for parameter estimates on a and the seasonal dummy variables.
equation suggests that this ARCH-like behavior in the residuals results from imposition of co-integration on this market. When we impose weak exogeneity on the Amarillo market (set all
a3j0; j1;. . .;6) the test on ®rst order ARCH residuals falls to 1.92, with an associated p -value of 0.165. However, we reject weak exogeneity (the zero restrictions on a) at a p-value of 0.000; suggesting that Amarillo is best not modeled as weakly exogenous). We did not take further steps to model ARCH-like eects in the Amarillo residuals.
Responses to perturbations in each of the long-run relations are given in the ®rst matrix on the right-hand side of Eq. (3); where perturbations in the long-run equilibrium are given byb0x(tÿ1), the second matrix and vector on the right-hand side of Eq. (3). We listt-statistics on alpha matrix entries in parentheses, which are relevant to the particular identifying restrictions imposed. In-terpretation of each alpha magnitude is interpretable, given the particular normalization used on each beta vector.
the next four cointegrating vectors. For our ®rst cointegrating vector normalization is with respect to the Kansas City rate in periodtÿ1:x1 tÿ1 0:64x2 tÿ1 ÿ3:79z1 tÿ1, wherez1 tÿ1 is used to represent perturbations in the ®rst long-run equilibrium and x1 tÿ1 and x2 tÿ1 represent the Kansas City and Dallas-Ft. Worth rates, respectively. When the Kansas City rate is high relative to its historical, long run, relation to the Dallas-Ft. Worth rate, the Kansas City rate falls in the subsequent period by 0.54z1 tÿ1. Similar interpretations exist for the response of the Kansas City rate for perturbations in the long-run relations between Dallas-Ft. Worth and Amarillo rates, Amarillo and Enid rates, Enid and Wichita rates, and Wichita and Salina rates. Interestingly, Kansas City does not respond signi®cantly to perturbations in the Salina±Denver long-run equilibrium.
Dallas-Ft. Worth shows a signi®cant negative response to disturbances in the second, third, fourth and ®fth long-run relations. For our second cointegrating vector the normalization is with respect to the Dallas-Ft. Worth market: x2 tÿ1 ÿ0:91x3 tÿ1 0:36z2 tÿ1, where
z2 tÿ1is used to represent perturbations in the second long-run equilibrium and x2 tÿ1 and
x3 tÿ1represent the Dallas-Ft. Worth and Amarillo rates, respectively. If the Dallas-Ft. Worth rate is high, relative to the long-run equilibrium (ifz2 tÿ1is positive) then the Dallas-Ft. Worth rate decreases by 0.79z2 tÿ1. Similarly, if the Amarillo rate is high in periodtÿ1, relative to the long-run equilibrium with Enid, x3 tÿ1 ÿ1:75x4 tÿ1 1:88z3 tÿ1, then the Dallas-Ft. Worth rate in periodt falls by 0.64z3 tÿ1. Similar interpretations exist for Dallas-Ft. WorthÕs response to the Enid±Wichita equilibrium and the Wichita±Salina equilibrium. Dallas-Ft. Worth does not respond signi®cantly to disturbances in the Salina±Denver equilibrium.
The Amarillo market responds signi®cantly negative with respect to shocks in the third long-run equilibrium: x3 tÿ1 ÿ1:75x4 tÿ1 1:88z3 tÿ1, so that when Amarillo is high rela-tive to its long-run relation with Enid in periodtÿ1, the Amarillo rate in the subsequent period falls by 0.54z3 tÿ1. Amarillo responds negatively, as well, to perturbations in the fourth long-run equilibrium:x4 tÿ1 ÿ4:13x5 tÿ1 8:73z4 tÿ1, so that when Enid is high relative to its long-run equilibrium with Wichita, the Amarillo rate responds negatively by 0.76z4 tÿ1. Similarly, Amarillo responds negatively with respect to the Wichita±Salina equilibrium.
Enid responds signi®cantly negative to perturbations in the fourth and ®fth cointegrating vectors (the Enid±Wichita and Wichita±Salina equilibria). Wichita shows only a weak negative response (t-statistic of 1.7) to shocks in the Salina±Denver equilibrium; so that when Salina is high relative to its long-run equilibrium with Denver, the rate in Wichita falls in the subsequent period. It is interesting that all other responses of the Wichita rate to shocks in long-run relations are not signi®cantly dierent from zero. Salina responds signi®cantly in a positive manner to shocks in the second, third, fourth and ®fth vectors and negatively to shocks in the last vector. Finally, Denver responds positively (signi®cantly so at usual levels) with respect to perturbations in all ®ve vectors.
4. Innovation accounting
The error correction model given in Eq. (3) can be used to summarize the period by period in¯uence each market rate has on the other rates of the seven variable system. The strengths of these relationships can be described through analysis of decompositions of forecast error variance (FEV decompositions). Critical to such an analysis is the method for treating contemporaneous innovation correlation. We follow the factorization commonly referred to as the ``Bernanke or-dering'' (see Doan, 1992). Write the innovation vector (ut) from the error correction model as:
Aut vt, where A is a 7´7 matrix and vt is a 7´1 vector of orthogonal shocks. A general
de-scription of the model considered is given in Eq. (4)
a11 a12 a13 a14 a15 a16 a17 obtained via the matrix A. A factorization is identi®ed (see Doan, 1992, 8±10), if there is no combination ofiandj i6jfor which both {aij} and {aji} are non-zero (here {aij} is elementi,j
of matrix A).
It was common in earlier VAR-type analyses to rely on a Choleski factorization, so that the A matrix is lower triangular, to achieve a just-identi®ed system in contemporaneous time. We apply directed graph algorithms given in Spirtes et al. (1993) to place zeroes on the A matrix. (A similar suggestion is made by Swanson and Granger (1997).) A directed graph is an assignment of causal ¯ow (or lack thereof) among a set of variables (vertices) based on observed correlation and partial correlation. Our six variable error correction model based on the identifying restrictions results in the following innovation correlation matrix (lower triangular entries only are printed in order:
Dx1, Dx2, Dx3, Dx4, Dx5; Dx6, and Dx7):
correlation between innovations in Dx1t and innovations in Dx2t given innovations in Dx5t we
would calculate the inverse of the following matrix (taking the corresponding elements fromV
above):
V1 1:00 0:11 1:00 0:10 0:29 1:00
2
4
3
5; V1
1:00
0:0851 1:00
0:0716 0:2821 1:00
2
4
3
5:
The matrixV1is the 3´3 matrix with lower triangular elementsV1;1,V2;1,V2;2,V5;1,V5;2; andV5;5 of the V matrix given above. The o-diagonal elements of the scaled inverse of theV1 matrix, denoted above by V1* , are the negatives of the partial correlation coecients between the cor-responding pair of variables given the remaining variables. So, for example, the partial correlation between innovations in rates from market one (Kansas City) and innovations in rates from market two (Dallas-Ft. Worth) given innovations in market ®ve (Wichita) is 0.0851. Under the assumption of multivariate normality, FisherÕszcan be applied to test for signi®cance from zero; where z q i;jjk;n 1=2 nÿ jkj ÿ31=2lnf j1q i;jjkj j1ÿq i;jjkjÿ1g and n is the number of observations used to estimate the correlations, q (i, j|k) is the population correlation between seriesiandj conditional on seriesk(removing the in¯uence of serieskon eachiand j), and |k| is the number of variables ink(that we condition on). Ifi,j, andkare normally distributed and r(i, j|k) is the sample conditional correlation of i and j given k, the distribution of z(q(i,
j|k),n)ÿz(r (i, j|k),n) is standard normal. In our case, the correlation between innovations in Kansas City and Dallas-Ft. Worth, given innovations in Wichita (0.0851), is not signi®cantly dierent from zero at 0.05 or 0.10 levels (actually the marginal signi®cance level is 0.449). So at usual levels of signi®cance (say 0.05 or 0.10) we conclude that innovations in the Kansas City rate and the Dallas-Ft. Worth rate are not related in contemporaneous time.
Directed graphs as given is Spirtes et al. (1993) provide an algorithm for removing edges be-tween markets and directing causal ¯ow of information bebe-tween markets. The algorithm begins with a complete undirected graph, where innovations from every market are connected with in-novations with every other market. We represent this complete undirected graph on inin-novations from the error correction model given in Eq. (3) in Fig. 3, panel a. The algorithm removes edges based on vanishing correlation and partial correlation, the latter measured based on the scaled inverse correlation matrix, as explained above. Edges between variables are removed sequentially based on either vanishing zero order correlation (unconditional correlations) or vanishing con-ditional correlations, where conditioning is done on all possible sets with members 1;2;. . .;Kÿ2, whereKis the number of variables studied.
Key to assigning direction of causal ¯ow between variables which remain connected after all possible conditional correlations have been passed as non-zero, is the notion of sepset. The conditioning variable (s) on removed edges between two variables is called the sepset of the variables whose edge has been removed (for vanishing zero order conditioning information (unconditional correlation) thesepsetis the empty set). Edges are directed by considering triples
X±Y±Z, such thatX and Yare adjacent as areY and Z, butXand Z are not adjacent. Direct edges between triplesX±Y±ZasX !Y ZifYis not in thesepsetofXandZ. IfX !Y,Yand
Z are adjacent,X and Z are not adjacent, and there is no arrowhead at Y, then orientY±Z as
Y !Z. If there is a directed path fromXtoY, and an edge betweenXandY, then orientX±Yas
Fig. 3 gives both the complete undirected graph and the ®nal directed graph on innovations from our seven market error correction model (Eq. 3). Panel A is the starting point from which edges are removed and edges directed according to the plan outlined above (actually according to the TETRAD II programs (Spirtes et al., 1993)). Panel B is the ending point. At the ®ve percent level, we have no directed edges. Innovations from the error correction model (Eq. 3) are inde-pendent in contemporaneous time. At a 10% signi®cance level we ®nd the directed edges as given in panel B. Applying a 10% signi®cance level we see edges running between Salina and Wichita to Dallas-Ft. Worth.
We test these restrictions using the likelihood ratio test for over identi®cation given in Doan, 1992. We conduct these tests on two alternative models. Our identi®cation restriction is that we can not have both aij60 and aji60 (actually this requirement is overly restrictive, but we
maintain it here, see Doan, 1992, pp. 8±10 for discussion). First, is the directed path found at 10%, where innovations from Wichita and Salina cause innovations in Dallas-Ft. Worth in
contemporaneous time. This path implies 19 zero restrictions (there are 21 lower triangular el-ements or their transpose elel-ements which can be non-zero in a just identi®ed model; the directed graph path found at 10% puts 19 of these equal to zero). These restrictions result in a chi squared statistic of 17.09. With 19 degrees of freedom, we reject these zero restrictions at a p -value of 0.58 (a rather high level), suggesting that the restrictions are consistent with the data. Our second test considers the pattern found at 5.0% (independence). Here we set all 21 lower triangular (and all upper triangular) terms equal to zero. These restrictions result in a chi squared statistic of 27.74; with 21 degrees of freedom, we reject the restrictions at the 0.15 signi®cance level. While independence of innovations in contemporaneous time are not rejected at usual (5%) signi®cance levels, clearly the large dierence in level of signi®cance [0.15±0.58] in tests, between the independence ordering and the Salina and Wichita causing Dallas-Ft. Worth ordering in contemporaneous time, suggest we ought to take the latter seriously. The error decompositions associated with this latter ordering of contemporaneous innovations are dis-cussed below.
Error decompositions associated with the error correction model are given in Table 3. Here decompositions are oered under the ordering of innovations in contemporaneous time generated by the directed graph at the 10% level. Responses for the 5.0% directed graph are similar to those oered here and may be obtained from the authors. (Further, responses from a VAR in levels give qualitatively the same results as those reported in Table 3. These too can be obtained from the authors at the address given at the beginning of the paper.)
Kansas City appears to be near exogeneous in the very short-run, as it explains over 95% of its own variation at horizons 0, 1, and 2 steps ahead. In the longer run, Kansas City rates are modestly explained by innovations in Salina (9.25%) and Dallas-Ft. Worth (4.17%). However, overall, and compared to other markets (Wichita excepted) Kansas City rates are not predomi-nately determined by rates in the other markets. Rates from Dallas-Ft. Worth are most aected (in the long run) by historical shocks from Salina and Wichita. Dallas-Ft. Worth accounts for about 50% of its own variation at the 12 month horizon with these other two markets accounting for in aggregate another 25%. Amarillo appears to be relatively more exogenous than the Dallas-Ft. Worth rate (as over two-thirds of its variation is accounted for by its own past innovations at the twelve horizon). At the long horizon of 12 months innovations in Salina and Dallas-Ft. Worth have the strongest in¯uence on the Amarillo rate.
Decompositions on the Enid market rate look similar to those on the Amarillo rate, about two-thirds of the variation in the Enid rate is accounted for by its own innovations at long horizons. Again innovations in the Salina rate are the next most important factor in the Enid rate; although contributions form the other four markets appear to be almost equally important.
5. Discussion
In this paper we consider the dynamics of export-wheat railroad rates in the central US plains. Our interest centered on rate dynamics, possible leadership in setting rates and interactions among markets in contemporaneous time and subsequent periods.
Table 3
Error decompositions on seven hinterland rail rates based on bernanke ordering of innovations from an error cor-rection model based on a directed graph model of innovations at the 10% level, 1988±1994 dataa
Steps ahead Std. error KC D-FW Amarillo Enid Wichita Salina Denver
(KC)
0 0.209 100.00 0.00 0.00 0.00 0.00 0.00 0.00
1 0.232 98.04 0.23 0.19 0.28 0.27 0.24 0.16
2 0.239 95.74 0.68 0.31 0.27 0.72 0.24 0.37
12 0.264 80.32 4.17 0.73 1.98 1.28 9.25 2.26
(DFW)
0 0.181 0.00 88.4 0.00 0.00 7.89 3.71 0.00
1 0.200 2.20 73.02 0.85 7.86 11.78 3.99 0.32
2 0.211 3.08 66.35 1.63 8.27 14.40 5.14 1.13
12 0.272 4.60 48.03 3.11 8.86 13.18 15.31 6.91
(Amarillo)
0 0.249 0.00 0.00 100.00 0.00 0.00 0.00 0.00
1 0.267 0.29 0.32 97.17 1.1 0.21 0.56 0.33
2 0.275 0.66 0.96 93.51 1.74 1.64 0.63 1.64
12 0.331 2.42 7.04 66.4 4.6 2.55 11.84 5.13
(Enid)
0 0.165 0.00 0.00 0.00 100.00 0.00 0.00 0.00
1 0.177 0.45 0.06 1.57 91.51 4.93 1.46 0.01
2 0.182 0.61 0.28 2.28 86.53 7.56 2.32 0.42
12 0.213 2.06 5.32 3.26 66.3 9.25 9.30 4.51
(Wichita)
0 0.099 0.00 0.00 0.00 0.00 100.00 0.00 0.00
1 0.117 0.00 0.01 0.52 0.00 97.23 0.69 1.53
2 0.125 0.07 0.10 0.94 0.03 95.19 1.18 2.48
12 0.132 0.91 1.00 1.47 0.23 90.6 1.89 3.90
(Salina)
0 0.123 0.00 0.00 0.00 0.00 0.00 100.00 0.00
1 0.152 0.42 4.85 0.08 2.26 0.37 88.98 3.04
2 0.174 0.33 8.83 0.50 4.71 0.28 80.42 4.92
12 0.315 3.41 17.14 3.26 9.80 3.29 51.65 11.15
(Denver)
0 0.105 0.00 0.00 0.00 0.00 0.00 0.00 100.00
1 0.128 2.32 14.99 0.30 0.51 0.42 1.27 80.20
2 0.142 4.68 18.33 0.63 1.47 0.68 5.14 69.06
12 0.237 6.10 20.48 2.94 7.59 3.32 25.57 33.99
a
The error decomposition analysis from an error correction model showed interesting rate dy-namics for the two Kansas regions and the Oklahoma region. Results show export rates in the Wichita region to be ``highly exogeneous''. Over 90% of the variation in Wichita export rates is accounted for by variations in its own past rates. Several factors may oer partial explanation. The market share of the leading carrier (ATSF) is estimated to be about 75%, the highest market share of any carrier in any region. Possibly the ATSFÕs comparatively large market share insulates this regionÕs export rates from those of other regions. The estimated market share held by the ATSF in the adjacent Oklahoma region (Enid) is also comparatively high (about a two-thirds market share) as is the exogeneity of export rates (66%).
The rate dynamics of the Wichita and Enid regions are in marked contrast to that of the Salina region, the other major shipping region. Export rates in the Salina region account for signi®cant variation in rates of all other regions except Wichita, and rates of three regions account for a portion of long-term variation in Salina rates. The waybill data show the Salina region to make a comparatively large portion of its export shipments from a few sites and to have a higher average number of cars per export shipment. This suggests carriers in north Kansas may have been more aggressive in providing incentives for shippers to consolidate for purposes of making more e-cient unit train shipments and abandoning nearby track to increase trac density. It was esti-mated that the UP was the primary carrier in the north Kansas (Salina) area with a market share that may have ranged as high as 65%; regardless, the UP would have encountered substantial competition from the ATSF in a portion of the region.
A study by Friedlaender et al. (1992) may oer some explanation: they found the UP to have been particularly successful in the deregulation era at rationalizing its cost structure. In contrast, the ATSF was comparatively poor at reducing costs through rail rationalization. The relatively high market share enjoyed by the ATSF in the Wichita and Enid region in combination with railroad management that was not aggressive regarding rationalization of its cost structure may have led to the observed ``highly exogeneous'' pricing dynamics. The pricing dynamics of the Wichita and Enid regions are in contrast to those of the Salina region, where the UP is the pri-mary carrier.
Results show the Salina regionÕs export rates to account for 26% of the variation in the Col-orado (Denver) study region export rates, the strongest inter-regional in¯uence identi®ed in the study. This is expected since the UP is the primary carrier in both regions and important con-solidated, unit train shippers in western Kansas are in close proximity to a unit train shipper in eastern Colorado, the principal wheat producing region in that state.
of the variation in the Salina and Denver region export rates. Thus, regions which tranship wheat via the Dallas-Ft. Worth region may ®nd it advantageous to adjust their direct export rates in reaction to Dallas-Ft. Worth export rates. This seems a reasonable rate reaction in view of the above described role of Dallas-Ft. Worth. In fact, it is curious that the other major shipping regions (south Kansas (Wichita), Oklahoma (Enid)) do not also react to Dallas-Ft. Worth rates. Again, the explanation may be a more aggressive railroad management in north Kansas (Salina) and Colorado (Denver), the only two regions where the UP is the primary carrier.
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